types of number

24 12 2009

Reading the story of e, final chapter, and several ideas spring to mind. Relating to the continuity of the number line, certain numbers being unusual, eg √2 or π.

1. Pythagorean obsession with musical intervals and ratios of string length… I would not be surprised to find out that the circularity of the cochlea has something to do with musical note interpretation. That is, it’s not the sound frequency that we apply musical sense to, but our sensation, the modulation of nerve signals derived from the physical massing of cells around cochlea.

2. The rational numbers are enough to describe most phenomena because

… the accuracy of any measurement is inherently limited by the accuracy of our measuring device

This parallels the quantum world where our instruments influence the thing measured. That is, e and π are irrational in that they do not have an absolute rational description m/n, they deny this measuring. We can only get as accurate as the tools we use. We settle on a number that is sufficient for our purpose. It is always an approximation.

3. √2 is irrational in that it can not be written m/n, it is not resolve nicely. And yet we can manipulate it. We can construct it easily with a unit square. We may not be able to specify it, but we can point at it. We can refer to it with the symbols √2, and we can point to it if we make a unit square and point at the length of the diagonal. But with our number system, we can not apply a discrete number to it. We can only ask that the mind look at a thing, or carry the consequence of an imaginary process, the square root of 2.

4. The author refers to irrational numbers, written in decimal form as non-terminating and non-repeating, as holes in the number continuum. I guess I can understand this in terms of there not being a n/m or a fixed pattern, and hence it is un-numbered. I take it a step further and think of them as being asymptotes.

5. Transcendental numbers are those that can be expressed as a solution to an algebraic equation. Hence, √2 is algebraic even though it is irrational. If it is not algebraic, it must be irrational; but some irrational numbers can be algebraic eg √2. A transcendental number can be expressed in terms of an infinite series of fractions. π and e are transcendental.

What follows is a little more basic, regarding primes.

6. The fundamental theorem of arithmetic is that any number above 1 can be factored into primes in one and only one way. No problem with this, though the number of primes in a given composite is interesting. eg 4 is 2×2, 8 is 2x2x2, so the shape of a number is given by the number of primes used to derive it. 12 is also a volumetric number since it is 2x2x3. Hmmm… 1 is 1x1x1x… 2 is 2x1x1x1…. 3 is 3x1x1x1… 4 is 2x2x1x1…. 5 is 5x1x1x1… 6 is 3x2x1x1….

7. Primes seem to arrange themselves into p and p+2 eg 3 and 5 or 11 and 13 or 17 and 19. Why do these prime twins exist? Primes can not follow one another, only 2 and 3 the only successive prime numbers. Surely this has to do with the number 2?

8. The Goldbach Conjecture: every even number greater than or equal to 4 can be written as a sum of two primes. It is unsolved. No counter-example has been provided, but no proof either. hmmm 0 as 0+0+0… 1 as 1+0+0+0… 2 as 1+1+0+0+0… 3 as 2+1+0+0+0… or 1+1+1+0+0+0…  4 as 3+1+0+0+0… or 2+2+0+0+0… or 2+1+1+0+0+0… or 1+1+1+1+0+0+0… etc I know I am including 0 and 1 in these calculations, but it makes some kind of sense since 0 is the identity of addition as 1 is the identity of multiplication. There are multiple ways of writing a prime sum, eg 10 can be written 5+5 or 3+7.

9 Lambert’s Conjecture: for large x, the number of primes below a certain number x ~ x/ln x, where ln is the natural log base e, otherwise known as the Prime Number Theorem. This is surprising to mathematicians, that the average distribution of prime numbers can be described using logarithmic function; that is, primes as the domain of integers and e to the domain of limits and continuity. But it should not be, since the number of higher dimensional shaped number increases as x gets larger. That is, 4 is the first square number, 8 is the first cube before we even get our second square number 9, and even by 16 we have our first hypercube number well before we get our third square number 25 and second cube number 27. Every order of 10 we increase x by, the number of powered shaped numbers will increase, won’t it? Of the number of primes within 100 million, which is 5,761,455, how many of them are power-derived? This might contribute the accuracy that this 5.75% happens to be close to 0.0543 which is 1/ln 100 million.

This all seems to be related to addition, the higher function of multiplication, and then powers. And because of the nature of large numbers, and the different ways of thinking of them as sums or products, the statistical behaviour of large primes seems reasonable enough. Also, primes are effectively the numbers that are not repetitions of previous numbers, ie not multiples. The reason why a formula is not possible is because it is the absence of pattern, rather than the presence of one; the primes are what is left. Still, e and π turn up as interesting numbers and not through exclusion of other numbers solutions in a set.




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